Environmental Engineering Reference
In-Depth Information
interaction. For geometric reasons, the interaction with the solvent is somewhat weaker
for inner sphere reactions, than for outer sphere ones. Besides the reactions shown in
Fig. 2.1, we shall also briefly discuss ion transfer, in which an ion passes from the outer
to the inner sphere position on the surface.
2.2 OUTER SPHERE ELECTRON TRANSFER
The theory of outer sphere electron transfer was first developed for homogeneous reac-
tions in polar solvents. A simple electron exchange between two partners in close
proximity was expected to be extremely fast, since the electron can tunnel rapidly
from one molecule to the other. Nevertheless, such electron exchange is comparatively
slow, and requires an energy of activation. The reason is that the electron transfer is
accompanied by a reorganization of the solvent: the initial and the final states of the
reactants carry different charges, and therefore their solvation, which is governed by
the interaction of the molecular charge with the dipole moment of the solvent, changes
during the reaction. Electron transfer is therefore preceded by a fluctuation of the
solvation to an intermediate state, and this requires an energy of activation. The key
concept of the theories of simple electron transfer is therefore the energy of reorgan-
ization of the solvent, which determines the energy required to reach this intermediate
state; its exact definition will be given below.
These ideas can be applied to electrochemical reactions, treating the electrode as
one of the reacting partners. There is, however, an important difference: electrodes
are electronic conductors and do not posses discrete electronic levels but
electronic bands. In particular, metal electrodes, to which we restrict our subsequent
treatment, have a wide band of states near the Fermi level. Thus, a model
Hamiltonian for electron transfer must contains terms for an electronic level on the
reactant, a band of states on the metal, and interaction terms. It can be conveniently
written in second quantized form, as was first proposed by one of the authors
[Schmickler, 1986]:
H
e
¼
1
a
n
a
þ
X
k
1
k
n
k
þ
X
k
V
k
c
k
c
a
þ
V
k
c
a
c
k
(2
:
1)
Here, n denotes a number operator, c
þ
a creation operator, c an annihilation operator,
and 1 an energy. The first term with the label a describes the reactant, the second term
describes the metal electrons, which are labeled by their quasi-momentum k, and the
last term accounts for electron exchange between the reactant and the metal; V
k
is
the corresponding matrix element. This part of the Hamiltonian is similar to that
of the Anderson - Newns model [Anderson, 1961; Newns, 1969], but without spin.
The neglect of spin is common in theories of outer sphere reactions, and is justified
by the comparatively weak electronic interaction, which ensures that only one electron
is transferred at a time. We shall consider spin when we treat catalytic reactions.
As discussed above, a crucial aspect is the interaction of the reactant with the
solvent. In a quantum theory, the solvent can be represented as a bath of harmonic
Search WWH ::
Custom Search